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In the figure below, π΄π΅ is a tangent to the circle π at π΅, πΆπ· is a diameter, the measure of angle π΅π΄π equals π₯, and the measure of angle ππ·π΅ equals two π₯ minus 55.
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Find the value of π₯ in degrees.
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Whenever you attempt this kind of question, I always say mark on diagram the angles that you already know or that you can work out first.
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Well, the first angle we know is that the angle π·π΅π is gonna be equal to two π₯ minus 55.
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And we know that it equals two π₯ minus 55 because itβs gonna be the same as angle ππ·π΅ because itβs an isosceles triangle.
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Well, how do we know that itβs an isosceles triangle?
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Well, itβs gotta be an isosceles triangle because π·π will have to equal ππ΅ because theyβre both radii of the circle.
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Okay, great, so now what else do we know?
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Well, the next angle we know is angle π΄π΅π.
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We know that angle π΄π΅π is equal to 90 degrees.
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So itβs a right angle and this is because itβs the angle between a tangent and a radius and the angle between tangent and radius is always 90 degrees.
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Okay, so now, weβve actually written all the angles that we know.
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So now, itβs actually time to actually calculate some additional angles.
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The first one weβre gonna start with is angle π·ππ΅.
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Well, we know that angle π·ππ΅ is gonna be equal to 180 minus two π₯ minus 55 minus two π₯ minus 55, which is gonna be equal to 180 plus 110 minus four π₯.
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Now, Iβll point out here a common mistake.
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The common mistake is actually to have had minus 110.
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And the reason you might have that is because obviously we could see that it was minus 55 minus 55.
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However, as itβs minus a minus, that makes it positive.
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So we get add 110.
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And we can say that angle π·ππ΅ is equal to 290 minus four π₯.
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And this is because the angles in a triangle add up to 180.
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So now, we can move on to angle π΅ππ΄, which is going to be equal to 180 minus then weβve got 290 minus four π₯.
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And thatβs because that was the angle π·ππ΅.
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And this gives us four π₯ minus 110.
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Again, be very careful here of the negative numbers cause we have got minus a negative, again which gives us positive four π₯.
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And again, we include our reasoning.
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And the reason for this is because the angles on a straight line equal 180 degrees.
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So therefore, angle π·ππ΅ and angle π΅ππ΄ must be equal to 180 degrees.
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Okay, great, so weβve now found that angle.
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So now, we can actually move on and find the angle weβre looking for, which is π΅π΄π.
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So therefore, we could say that the angle π΅π΄π is equal to 180 minus 90 minus four π₯ minus 110.
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Well, therefore, we can say that π₯ is equal to 200 minus four π₯.
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And we can say π₯ because we can see that angle π΅π΄π is equal to π₯.
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So now, weβve got π₯ is equal to 200 minus four π₯ and we can solve to find π₯.
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So then if we add four π₯ to each side, we get five π₯ equals 200.
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And then we divide both sides by five.
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We get π₯ equals 40.
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So therefore, we can say that the measure of angle π΅π΄π is equal to 40 degrees.
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And therefore, π₯ is equal to 40 degrees.
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Okay, great, so weβve got to our final answer.
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But what we want to do here is actually quickly double check it.
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Iβm going to check it by using the first triangle, which is triangle π΅π·π.
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So we have all three angles which are two π₯ minus 55 plus two π₯ minus 55 plus 290 minus four π₯.
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So now, if I actually substitute in our value for π₯ which is 40, weβre gonna get 80 minus 55 because two multiplied by 40 gives us 80 plus another 80 minus 55 minus 290 minus 160 because again four multiplied by 40 equals 160.
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Okay, great, so letβs calculate this.
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Well, this gives us 180.
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Well, we know that this is correct because the angles in a triangle add up to 180.
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So therefore, yes, we can say with confidence that π₯ is equal to 40 degrees.